1. P ROBLEM OF THE D AY Directions : 1. Complete the problem of the day working alone or with a partner. After you have completed the task, identify at least one indicator from the mathematical practices that is evident in your work. 2. Look at the student work that is based on the same problem and identify at least three indicators from the mathematical practices. Be prepared to share your findings with the rest of the group. 1

2. PROBLEM OF THE DAY Problem of the Day: RATIO PROBLEM. Monica is 16 years old and her brother Sam is 4 years old. 1. Write 5 different ways to compare Monica's and Sam's age. 2. For each of the 5 ways, describe for your audience why each way is communicating the same idea. 3. To help your audience understand more clearly, draw diagrams to accompany some of your explanation your explanation. Reflection - Was this a challenging task for you? If so, why? If not, why not? - What information have you learned that helped you to solve this problem? - Name one situation that you have identified outside of the classroom that you used or can use your understanding of ratio to solve. 2

3. STRUCTURES AND FRONT-LOADING STRUCTURES It is the beginning of the school year. You are expecting a new class of students. What do you have in your basket to begin the school year? To begin the school year, you have to fill the first of 10 empty baskets. The other baskets will be filled during the school year. On the colored strips, write as many things that will go into basket # 1. Draw a picture of your basket on chart paper and glue your strips onto the basket then display your basket around the room. 3

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5. S TRUCTURES AND F RONT -L OADING ? Some of the most important pieces of collections in your basket are students’ learning goals. These learning goals are defined by the two pillars of education: “What students should know?”, and “What students should be able to do?” On these two pillars rest all the decisions that govern curriculum and instruction. According to Michael Fullan (2008), highly effective schools leave nothing important to chance. This can also be translated to mean that a highly successful teacher does not leave the education of her/his student to chance. How you decide students’ learning goals are dependent on how deeply you are examining the collections in your basket and what exactly you are finding. 5

6. S TRUCTURES AND F RONT -L OADING ? Learning goals, according to Tim Westerberg (2009), state what students will understand or be able to do as a result of engaging in one or more well-constructed learning activities. Learning activities ( preparing a report ); performance goals (improving two reading levels); are different from learning goals ( understand the relationship between predator and prey). The research, according to Westerberg (2009), has shown that when clear learning goals are established and communicated to students early and often, this help students to seek challenges, show persistence in the face of adversity, use more effective learning strategies, have more positive attitudes, and are more cognitively engaged in learning. 6

7. S TRUCTURES AND F RONT -L OADING ? Today’s Goal : The purpose of today’s workshop is for you to identify a goal based on a specific reason then begin to create a plan for instruction, assessment, then looking at results in order to do adjust your instructional goals if you do not get the desired results. I am a 6 grade math teacher and in my basket I found the End-of-Year assessment for my students. A large percent of my students appeared to have difficulty with ratio and proportion. I looked at and compared the NYS Standards to the CCSS that address ratio. * 7

8. STRUCTURES AND FRONT-LOADING 8 CCSS 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 2. Understand the concept of a unit rate a/b associated with a ratio a : b with b  0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” 26 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. 6.N.6 – Understand the concept of rate 6.N.7 – Express equivalent ratios as a proportion 6.N.8 – Distinguish the difference between rate and ratio 6.N.9 – Solve proportions using equivalent fractions 6.N.10 – Verify the proportionality using the product of the means equals the product of the extremes 6.N.11 – Read, write, and identify percents of a whole (0% t0 100%) 6.N.12 – Solve percent problems involving percent, rate, and base.

9. S TRUCTURES AND F RONT -L OADING ? Using the UBD Framework, we are going to plan a unit on Ratio and Proportion. 9 Unit Topic: Ratio and Proportional Relationships Established Goal: Understand the ratio concepts and use ratio reasoning to solve problems. Understandings: Ratios are used to compare two or more quantities Ratios differ from fractions Ratios do not follow fraction rules Ratios are ordered pairs A proportion is an equation showing that two ratios are equal When two quantities are proportional, a change in one quantity corresponds to a predictable change in the other. Rate is a special kind of ratio

10. S TRUCTURES AND F RONT -L OADING ? Essential Question : If ratios can be written as fractions but do not follow the fraction rules, what is unique about ratios? Beside numbers, does natural proportional relationships exist? 10

11. S TRUCTURES AND F RONT -L OADING ? 11 Students will know : Equal ratios form proportion A ratio is the quotient of two numbers A ratio is an ordered pair To find ratio of two or more quantities, all quantities must be expressed in the same unit of measure. The denominators of fractions and ratios are chosen differently When both terms of a ratio are multiplied or divided by the same non-zero number, the results are the terms of another ratio equivalent to the original ratio.

12. S TRUCTURES AND F RONT -L OADING ? Students will be able to : Clearly state the difference between a fraction and a ratio. Solve problems using equivalent ratios Find the ratios of two quantities that are expressed in different units of measure Solve unit rate problems including those involving unit pricing and constant speed. Create proportions in order to solve problems 12

13. S TRUCTURES AND F RONT -L OADING ? Performance Tasks : What to consider? Put yourself in the position of the student attempting a test question or a performance assessment task. Ask yourself: -How would I have to think to answer this question? (assess thinking skills) - What do I have to think about to answer the question or do the task (assess content- knowledge) Students should be assessed on thing they were taught to do, not surprised on a test or performance with task for which they have no practice. 13

14. S TRUCTURES AND F RONT -L OADING ? What are Performance Tasks ? Complex challenges that mirror the issues and problems faced by adults. Ranging in length from short-term tasks to long-term, multi-staged projects, they yield one or more tangible products and performances. - The setting is real or simulated - Typically require the student to address an unidentified audience real or imagines - Are based on a specific purpose that relates to the audience -Allow students greater opportunity for personalization 14

15. S TRUCTURES AND F RONT -L OADING ? Performance Tasks : Get tubes of primary and secondary colors from the art room. Develop a ratio chart and produce three different shades of a (color of your choice). - Share the ratio chart with a friend and the colors without identifying which sets of ratios produce which shade. See if your friend can identify which ratio of two or three colors produced which shade? - Describe something new that you have learned from this activity? - What one conclusion you can draw about the usefulness of ratios. 15

16. S TRUCTURES AND F RONT -L OADING ? Are you curious about your body parts. Do you think they are proportional? Design and carry out tasks that can help to satisfy your curiosity. Other Evidence : Quizzes Student self-assessment During unit test Student reflection End of unit test Exit slips Quick writes Journal entries 16

17. S TRUCTURES AND F RONT -L OADING ? Homework : Create the Performance Assessment for the grade you worked on. Come prepared to share your work through a gallery walk. Next weeks lesson will look at lesson planning for the unit. 17